Foliations on non-metrisable manifolds: Absorption by a Cantor black hole

Abstract

We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. It is shown that Kneser's pathology of a manifold foliated by a single leaf cannot occur with foliations of dimension-one. On the other hand, there are open surfaces admitting no foliations. This is derived from a qualitative study of foliations defined on the long tube S 1 × L + (product of the circle with the long ray), which is reminiscent of a 'black hole', in as much as the leaves of such a foliation are strongly inclined to fall into the hole in a purely vertical way. More generally the same qualitative behaviour occurs for dimension-one foliations on M × L + , provided that the manifold M is "sufficiently small", a technical condition satisfied by all metrisable manifolds. We also analyse the structure of foliations on the other of the two simplest long pipes of Nyikos, the punctured long plane. We are able to conclude that the long plane L 2 has only two foliations up to homeomorphism and six up to isotopy.

Some steps on short bridges: non-metrizable surfaces and CW-complexes

“…This holds too for maps from ω 1 to a finite union of Euclidean open subsets, for instance by embedding them is some R n . (In fact, any continuous map ω 1 → X with X first countable, Hausdorff and Lindelöf is eventually constant, see Lemma 4.3 in [2].) We now follow the proof of Proposition 7.1 in [1].…”

Actions of groups of foliated homeomorphisms on spaces of leaves

“…The spaces of leaves Y of foliations often appear as spaces of orbits of flows and more generally of group actions and play an important role in the understanding the dynamics of that actions, e.g. [5,15,4,7,10,2,3] and others. The usual difficulty arising at once when we pass from the manifold X to the space of leaves Y is that Y is usually non-Hausdorff.…”

Dynamics of non-metric manifolds

“…Thus, the Poincaré-Bendixson theory-relying on the sack argument acting as a trap for trajectories on a surface where Jordan separation holds true-also propagates non-metrically. From it and Schoenflies, one can draw a hairy ball theorem for ω-bounded 4 simply-connected surfaces, yielding a wide extension of the fact that the 2-sphere cannot be brushed. (By a brush, we shall mean 1 To quote again-this time loosely-Poincaré [60, p. 82].…”

“…and in the noncompact case Kerékjártó 1923 [40]. 4 Recall that a ω-bounded space is one such that any countable subset admits a compact closure. This point-set concept when particularised to manifolds allows one to hope recovering some of the "finistic" virtues of compact manifolds beyond the metric realm, cf.…”

“…• Two islands delineated by frames, packaging results of previous papers by the authors. Since the present paper emphasises the viewpoint of flows, the paper [4] (concerned with foliations) is not an absolute prerequisite. The note [23] (Jordan and Schoenflies in non-metrical analysis situs) will be used in some arguments.…”